1.4.2 Curvature (round 1)
The unit tangent changes rapidly at portions of a path that are very curvy and changes much less through straighter portions of a the path. This is why mathematicians have chosen to use the unit tangent to quantify the "curviness" of a curve.
We want to measure how quickly is changing which suggests using a derivative.
is a vector. To get a scalar quantity we might consider .
At first this quantity has some good reasons to recommend it. If you experiment with the applet below you'll see that circles have constant curvature and line segments have zero curvature as one might expect. However one flaw with using this quantity to assess curvature is that it depends on parameterization. In the applet below you can change the base path but you can also change the parameterization by changing . You'll see that the value of will change simply by changing the parameterization.